1st person hand shakes with 2,3,4 is equal to 3 handshakes
2nd person hand shakes with 3,4 is equal to 2 handshakes
3rd person hand shakes with 4 is equal to 1 handshake
4th person has already shook hands with all of them, so a total of 6

People the answer is 6.
Here's a clear explanation.
There are 4 people namely A B C D.
A shakes hand with B C D.
B shakes hand with C D. (B has already shook hands with A in the line above so he won't do it again)
C shakes hand with D.
It's essential to remember that everyone shakes hands with others only once. So AB is the same as BA.
I hope it's clear.
If not I'll be happy to answer any query.